Geodesic
Minimal proper quasifields with additional conditions
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 1, pp. 104-113

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We investigate the finite semifields which are distributive quasifields, and finite near-fields which are associative quasifields. A quasifield $Q$ is said to be a minimal proper quasifield if any of its sub-quasifield $H\ne Q$ is a subfield. It turns out that there exists a minimal proper near-field such that its multiplicative group is a Miller–Moreno group. We obtain an algorithm for constructing a minimal proper near-field with the number of maximal subfields greater than fixed natural number. Thus, we find the answer to the question: Does there exist an integer $N$ such that the number of maximal subfields in arbitrary finite near-field is less than $N$? We prove that any semifield of order $p^4$ ($p$ be prime) is a minimal proper semifield.
Keywords: quasifield, semifield, near-field, subfield. 
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     author = {Olga V. Kravtsova},
     title = {Minimal proper quasifields with additional conditions},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {104--113},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2020_13_1_a9/}
}
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Olga V. Kravtsova. Minimal proper quasifields with additional conditions. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 1, pp. 104-113. http://geodesic.mathdoc.fr/item/JSFU_2020_13_1_a9/